Question

Testing Claims About Proportions. In Exercises 9–32, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.

Overtime Rule in Football Before the overtime rule in the National Football League was changed in 2011, among 460 overtime games, 252 were won by the team that won the coin toss at the beginning of overtime. Using a 0.05 significance level, test the claim that the coin toss is fair in the sense that neither team has an advantage by winning it. Does the coin toss appear to be fair?

Short answer

Null hypothesis: The proportion of overtime wins that arewon by the team that has won the coin toss is equal to 0.5.

Alternative hypothesis: The proportion of overtime wins that arewon by the team that has won the coin toss is not equal to 0.5.

Test statistic: 2.05

Critical value: 1.96

P-value: 0.0404

The null hypothesis is rejected.

There is enough evidence to reject the claim that the proportion of games that arewon by the team that has won the coin toss is equal to 0.5.

The coin toss does not appear to be fair.

Step-by-step solution

Given information

Out of 460 overtime games, 252 of them arewon by the team that has won the coin toss. It is claimed that the coin toss is fair, and an equal number of teams who win/do not win the coin toss go on to win the game.

Hypotheses

The null hypothesis is written as follows.

The proportion of overtime wins that arewon by the team that has won the coin toss is equal to 0.5.

\({H_0}:p = 0.5\).

The alternative hypothesis is written as follows.

The proportion of overtime wins that arewon by the team that has won the coin toss is not equal to 0.5.

\[{H_1}:p \ne 0.5\].

The test is two-tailed.

Sample size, sample proportion,and population proportion

The sample size is equal to n=460.

The sample proportion of overtime wins that arewon by the team that has won the coin toss is computed below.

\[\begin{array}{c}\hat p = \frac{{252}}{{460}}\\ = 0.548\end{array}\].

The population proportion of games that arewon by the team that has won the coin toss is equal to 0.5.

Test statistic

The value of the test statistic is computed below.

\(\begin{array}{c}z = \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\ = \frac{{0.548 - 0.5}}{{\sqrt {\frac{{0.5\left( {1 - 0.5} \right)}}{{460}}} }}\\ = 2.05\end{array}\).

Thus, z=2.05.

Critical value and p-value

Referring to the standard normal table, the critical value of z at\(\alpha = 0.05\)for a two-tailed test is equal to 1.96.

Referring to the standard normal table, the p-value for the test statistic value of 2.05 is equal to 0.0404.

Asthe p-value is less than 0.05, the null hypothesis is rejected.

Conclusion of the test

There is enough evidence to reject the claim that the proportion of games that arewon by the team that has won the coin toss is equal to 0.5.

This concludes that the coin toss is not fair, and there is a greater chance of winning the game after winning the coin toss.